Error threshold in optimal coding, numerical criteria and classes of universalities for complexity

The free energy of the Random Energy Model at the transition point between ferromagnetic and spin glass phases is calculated. At this point, equivalent to the decoding error threshold in optimal codes, free energy has finite size corrections proportional to the square root of the number of degrees. The response of the magnetization to the ferromagnetic couplings is maximal at the values of magnetization equal to half. We give several criteria of complexity and define different universality classes. According to our classification, at the lowest class of complexity are random graph, Markov Models and Hidden Markov Models. At the next level is Sherrington-Kirkpatrick spin glass, connected with neuron-network models. On a higher level are critical theories, spin glass phase of Random Energy Model, percolation, self organized criticality (SOC). The top level class involves HOT design, error threshold in optimal coding, language, and, maybe, financial market. Alive systems are also related with the last class. A concept of anti-resonance is suggested for the complex systems.Comment: 17 page

Electrostatically-induced modal crosstalk phenomena in resonant MEMS sensors

International audienceThe aim of this paper is to introduce an explanation for the amplitude saturation and inhibition phenomena in resonant MEMS pressure sensors, via the electrostatic coupling of two resonance modes. Our analysis and experimental results reveal that these phenomena may be ubiquitous in electrostatic resonant MEMS sensors

Nonlinear resonances of electrostatically actuated nano-beam

Nonlinear response of electrostatically actuated nano-beam near-half natural frequency is studied by considering the nonlinearities of the large deformation, electrostatic force and Casimir effect. A first-order fringe correction of the electrostatic force, large deformation, viscous damping, and Casimir effect are included in the dynamic model. The dynamics of the resonator are investigated by using the method of multiple scales in a direct approach to the problem. The sufficient conditions of guaranteeing the system stability and a saddle-node bifurcation are studied. The influences of large deformation, damping, actuation, and fringe effect on the resonator response are studied. The peak amplitude of the primary resonance is given in the paper. Numerical simulations are conducted for uniform nano-beam

Nonlinear Forced Vibration of Piezoelectric and Electrostatically Actuated Nano/Micro Piezoelectric Beam

In this study, the nonlinear vibration analysis of nano/micro electromechanical (NEMS/MEMS) piezoelectric beam exposed to simultaneous electrostatic and piezoelectric actuation. NEMS/MEMS beam actuate with combined DC and AC electrostatic actuation on the through two upper and lower electrodes. An axial force proportional to the applied DC voltage is produced by piezoelectric layers present via a DC electric voltage applied in the direction of the height of the piezoelectric layers. The governing differential equation of the motion is derived using Hamiltonian principle based on the Eulere-Bernoilli hypothesis and then this partial differential equation (PDE) problem is simplified into an ordinary differential equation (ODE) problem by using the Galerkin approach. Hamiltonian approach has been used to solve the problem and introduce a design strategy. Phase plane diagram of piezoelectric and electrostatically actuated beam has plotted to show the stability of presented nonlinear system and natural frequencies are calculated to use for resonator design. The result compare with the numerical results (fourth-order Runge-Kutta method), and approximate is more acceptable and results show that one could obtain a predesign strategy by prediction of effects of mechanical properties and electrical coefficients on the stability and forced vibration of common electrostatically actuated micro beam

A parametric electrostatic resonator using repulsive force

In this paper, parametric excitation of a repulsive force electrostatic resonator is studied. A theoretical model is developed and validated by experimental data. A correspondence of the model to Mathieu\u27s Equation is made to prove the existence and location of parametric resonance. The repulsive force creates a combined response that shows parametric and subharmonic resonance when driven at twice its natural frequency. The resonator can achieve large amplitudes of almost 24 μm and can remain dynamically stable while tapping on the electrode. Because the pull-in instability is eliminated, the beam bounces off after impact instead of sticking to the electrode. This creates larger, stable trajectories that would not be possible with traditional electrostatic actuation. A large dynamic range is attractive for MEMS resonators that require a large signal-to-noise ratio

Bifurcation Type Change of AC Electrostatically Actuated MEMS Resonators due to DC Bias

This paper investigates the nonlinear response of microelectromechanical system (MEMS) cantilever resonator electrostatically actuated by applying a soft alternating current (AC) voltage and an even softer direct current (DC) voltage between the resonators and a parallel fixed ground plate. The AC frequency is near natural frequency. This drives the resonator into nonlinear parametric resonance. The method ofmultiple scales (MMS) is used to solve the dimensionless differential equation of motion of the resonator and find the steady-state solutions.The reduced order model (ROM) method is used to validate the results obtained using MMS. The effect of the soft DC voltage (bias) component on the frequency response is reported. It is shown that the DC bias changes the subcritical Hopf bifurcation into a cyclic fold bifurcation and shifts the bifurcation point (where the system loses stability) to lower frequencies and larger amplitudes

Dynamics of a non-linearly damped microresonator under parametric excitation and its application in developing sensitive inertial sensors with ultra-wide dynamic ranges

We model and investigate the response of a nonlinear cantilever beam under principal parametric excitation. The design is initially assessed, optimized, and tuned using three-dimensional finite element analysis (FEA) to ensure the presence of fundamental parametric resonance and the absence of other internal and higher-order parametric resonances. The derived governing differential equation represents a modified generalized parametrically excited dynamic system under principal parametric excitation. The nonlinear dynamic system is developed and presented in the context of resonators with extensive applications in developing sensors, filters, and switches. The quadratic and cubic nonlinearities include second- and third-order deflection, velocity, acceleration terms describing stiffness, damping, and inertial nonlinearities. To explore and investigate the corresponding generalized nonlinear Mathieu equation, the method of multiple-scales along with the reconstitution method are used and modulation equations are obtained and solved to obtain closed-form amplitude and phase equations. The quadratic damping is modeled and approximated using a Fourier series and analytical models are generated in both Cartesian and Polar frames. To further explore the dynamic system and its applications, a resonator is designed to measure external acceleration and investigated for two cases. It is discussed and shown how the external acceleration modifies the dynamic system, the corresponding reduced-order model, and the modulation equations. The external acceleration affects the amplitude, phase, and frequency of oscillation providing means to estimate the input. These results indicate that the proposed resonator design (dynamic system) is able to significantly improve the dynamic range of shock/acceleration sensors

Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion

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